This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 128.112.200.107

This content was downloaded on 14/12/2013 at 20:18

Please note that terms and conditions apply.

Tactile interactions lead to coherent motion and enhanced chemotaxis of migrating cells

View the table of contents for this issue, or go to the journal homepage for more

2013 Phys. Biol. 10 046002

(http://iopscience.iop.org/1478-3975/10/4/046002)

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING PHYSICAL BIOLOGY

Phys. Biol. 10 (2013) 046002 (8pp) doi:10.1088/1478-3975/10/4/046002

Tactile interactions lead to coherentmotion and enhanced chemotaxis ofmigrating cellsLuke Coburn1,5, Luca Cerone2, Colin Torney3, Iain D Couzin3

and Zoltan Neufeld4

1 School of Mathematical Sciences and Complex and Adaptive Systems Laboratory,University College Dublin, Dublin, Ireland2 European Bioinformatics Institute (EMBL-EBI), Wellcome Trust Genome Campus,Cambridge CB10 1SD, UK3 Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA4 School of Mathematics and Physics, University of Queensland, Brisbane, Australia

E-mail: lukecoburn@gmail.com

Received 3 January 2013Accepted for publication 3 May 2013Published 11 June 2013Online at stacks.iop.org/PhysBio/10/046002

AbstractWhen motile cells come into contact with one another their motion is often considerablyaltered. In a process termed contact inhibition of locomotion (CIL) cells reshape and redirecttheir movement as a result of cell–cell contact. Here we describe a mathematical model thatdemonstrates that CIL alone is sufficient to produce coherent, collective cell migration. Ourmodel illustrates a possible mechanism behind collective cell migration that is observed, forexample, in neural crest cells during development, and in metastasizing cancer cells. Weanalyse the effects of varying cell density and shape on the alignment patterns produced andthe transition to coherent motion. Finally, we demonstrate that this process may have importantfunctional consequences by enhancing the accuracy and robustness of the chemotacticresponse, and factors such as cell shape and cell density are more significant determinants ofmigration accuracy than the individual capacity to detect environmental gradients.

S Online supplementary data available from stacks.iop.org/PhysBio/10/046002/mmedia

1. Introduction

Collective migration of cells plays an important role inembryological development, tissue repair and cancer invasion[1–4]. Some migrating cells move as tightly connected clustersor sheets, while others form loosely associated streams inwhich cells make occasional temporary contacts. One classicalexample of such collective migration is produced by a groupof highly mobile undifferentiated cells called neural crestcells that migrate through the developing embryo to distantlocations and subsequently give rise to many different typesof tissues [5, 6]. Multicellular organization is also common intissue invasion by malignant tumour cells [7]. Targeting such

5 Author to whom any correspondence should be addressed.

collective behaviour may be effective in controlling the spreadof cancer [8]. Despite these widespread implications, themechanisms by which large groups of cells can self-organizefrom local cell–cell interactions remains poorly understood.

Collective migration has been studied extensively inthe context of animal groups, such as flocking birds, insectswarms and fish schools. Theoretical models and computersimulations of animal migration have shown that the generalmechanisms responsible for the emergent group levelmigratory properties can be captured by simple interactionrules such as attraction, repulsion and local alignment[9–13]. One of the key differences between these migrationsand collective cell migration is that cells are restricted tohighly localized information about their neighbours. As largerorganisms, such as birds and fish, use sight to detect and

1478-3975/13/046002+08$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

Phys. Biol. 10 (2013) 046002 L Coburn et al

interact with neighbours at a distance, simulations of theirmotion often consider these animals as point particles thatinteract via non-local interaction rules. However, the onlyinformation available to cells about the location and behaviourof their neighbours comes either in the form of diffusedchemical signals, that in most cases cannot provide accuratedirectional information, or from direct physical contact withother cells. Therefore to model realistic cell–cell interactionsit is necessary to take into account the spatial extensionand dynamically changing shape of the cells that cannot bedescribed by simple point particle models.

The cytoskeleton of cells is an elaborate network ofbiopolymers which can readily reorient itself into a newshape. Cells move via extensions of their cytoskeletons calledprotrusions which attach to the substrate while contractileforces within the cell cytoskeleton drag it forward in thedirection of the protrusions [14]. These protrusions also sensethe environment immediately surrounding the cell. Cellularprotrusions are unstable, forming and collapsing constantlyin every direction. After formation a protrusion may eithercollapse or branch to produce further protrusions [15, 16].Cells in motion are polarized with the protrusions on oneend more stable than the other and as such the time-averageddistribution of protrusions has a generally asymmetric shape.The stability and distribution of protrusions around thecell body may be influenced by external chemical signalsproducing directional movement, called chemotaxis [16, 17].

The movement of cells is affected by interactions withother cells. In the case of dense confluent cell sheets suchinteractions primarily involve mechanical stresses due toadhesion, volume exclusion and proliferation [18–21] thatcan described for example by the so-called cellular Pottsmodel (CPM) [22]. The CPM has been extended recentlyto represent active cell motility [23, 24]. Kabla has shownrecently [24] that a model in which the mechanical interactionsof the CPM are complemented with motile forces generatedby polarized cells can reproduce a variety of collectivesheet migration pattern observed in experiment. In looselyassociated stream-like migration cell protrusions transmitbiochemical signals. For neural crest cells, whose migrationshares many characteristics with metastatic cancer cells,such interactions provide directional guidance that leadsto coordinated migration even in the absence of externalchemoattractant [6, 25–27]. When two cells touch a chemicalexchange occurs and in response to this signalling, theprotrusions at the site of contact collapse. As the protrusionsare responsible for the motion, this causes the cells to redirectand after some time the cell reforms the lost protrusions ina new position on the cell surface while the cytoskeletonreorients itself. This type of contact inhibition of locomotion(CIL) was first observed by Abercrombie in tissue cultures[28] and loss of CIL has been proposed to contribute tothe invasiveness of malignant cells [29]. Recent in vivoexperiments clearly demonstrate that CIL plays a key rolein the directional migration of neural crest cells [26, 30, 31].Furthermore, in the presence of an external chemoattractantCIL was shown to promote collective chemotaxis of neuralcrest cells. While separated cells are unable to move in the

direction of gradient without interacting with other cells, theefficiency of collective migration of a cell population towardsthe chemoattractant was found to be strongly enhanced byincreasing cell density [32, 33].

Here, we use a computational model of CIL, based on theexperimental observations described above, to investigate themechanisms responsible for the emergent collective migrationof cells. Exploiting similarities to models of animal migration,we analyse quantitatively the conditions and characteristicsof collective migration of cell populations. We investigatethe relationship between the statistics of pairwise interactionsand the overall group migration, and study the influenceof the highly localized tactile interactions on chemotaxis todetermine the relative role of individual chemotactic abilitiesversus cell–cell social interactions. We note, that the aim ofthis model is not to reproduce the migration of a particular celltype in a specific biological context, but to analyse the role ofone particular type of interaction, CIL, in contributingto collective migration while we neglect other type ofinteractions, e.g. chemo-attraction and cell–cell adhesion.

2. The model

We consider a computational model of cell migration in aperiodic domain based on a simple tractable interpretationof CIL. In this model cells are composed of a circularcore surrounded by a deformable region representing cellprotrusions. The overall cell shapes are modelled as closedcontours in two dimensions. All cells in our simulations arepolarized. The freely moving cell has a front–back asymmetrywith protrusions concentrated at the front. We initially definethe shape of the cell in polar coordinates with a Gaussianfunction of the angle θ as:

P(θ; θ0) = R0 + Ae−(θ−θ0 )2/2ω2, θ0 − π < θ < θ0 + π

(1)

where R0 is the radius of the core of the cell, θ0 is theinstantaneous cell orientation and ω and A are cell shapeparameters. In the numerical simulations the contour isrepresented by a discrete set of m points corresponding toequally distributed orientations: θk = 2πk/m, k = 1, . . . , m.We use m = 50 in our simulations. Figures 1(a)–(c) showsexamples of cell shapes corresponding to different parametervalues. The cell shape parameters ω and A are varied such thatthe cell area remains constant.

To relate the motion of the cell to its shape, we assumethat the protrusions impart a radial force on the cell in thedirection of their growth proportionally to their length. Thuswe define the velocity of a cell to be proportional with theresultant contributions from protrusions in all directions aboutthe centre of the cell:

v(t) = v0

∫ θ0+π

θ0−π

(P(θ ) − R0)nθ dθ (2)

where nθ is the radial unit vector in the direction θ and theprefactor v0 is a constant. Note that the cell shape affects cellspeed and speed increases as ω decreases and A increases. Inthe absence of contact interactions a single cell moves with a

2

Phys. Biol. 10 (2013) 046002 L Coburn et al

(a)

(d )

(e)

(b) (c)

Figure 1. (a)–(c) Varying ω and A yields different cell shapes (each with θ0 = π

2 ). (a) ω = 0.3 and A = 1.0. (b) ω = 0.9 and A = 0.44.(c) ω = 1.5 and A = 0.3. (d), (e) Examples of pairwise collisions: when two cells touch a portion of their protrusions overlap. CIL causes thecells to lose protrusions and, as a result, the cells redirect their motion. The cell shape is restored symmetrically about the new angle ofmotion. The red arrows show the cells’ velocity. (d) Glancing collision. (e) Head on collision.

constant velocity and its orientation θ0 changes stochasticallyaccording to a Wiener process:

dθ0(t)

dt= η(t), 〈η(t)η(t − τ )〉 = σ 2δ(τ ), (3)

where η(t) is an uncorrelated white noise and σ is the noiseintensity. A target Gaussian shape function P0(θ; θ0(t)) is thendefined with the actual orientation using (1) and the cell shaperelaxes towards this new Gaussian contour according to:

∂P(θ, t)

∂t= −γ [P(θ, t) − P0(θ; θ0(t))] (4)

where γ determines the rate of regrowth and θ0(t) is theinstantaneous direction of motion at time t.

When two cells meet the protrusions are withdrawnwithin the area of contact, resulting in a reorientation of thedirection of motion. In the computational implementation,overlapping segments of the cells are detected and in thecorresponding radial directions the contours are reset to thecore radius. The collapse of protrusions breaks the symmetryof the cell with respect to the original orientation axis andchanges the direction of motion. The loss of protrusionsduring cell contact typically leads to a temporary decreasein the speed of locomotion. Following the reorientation dueto the modified shape of the contour the cells move awayfrom each other and gradually regain their normal shape andspeed by reforming protrusions symmetrically along the newdirection of movement (figures 1(d) and (e)). In the numericalsimulations the velocity vectors and cell contours are updatedusing discrete time steps. At each step first the velocity

vector is determined from the newly updated cell shape P(θ )

according to (2), and the orientation of the velocity vector iscombined additively with the noise corresponding to (3). Theresulting direction θ0(t) determines the orientation of the targetGaussian cell shape P0(θ0(t)) in the relaxation dynamics (4).

We note that real cell protrusions are typically moreirregular and dynamically changing [15, 16]. Recent workby Vedel et al [34] presents a similar computational model ofcells migrating that interact via CIL. In their model multipleshort-lived protrusions form in spikes about the cell body. Thecontour defined by (1) can be considered as a time-averagedrepresentation of the distribution of such random extensionsaround the cell.

3. Results

3.1. Collective migration

Using our cell interaction model we performed simulations ofgroups of interacting cells. To avoid dispersion and maintaina constant density, we assume that the cells are restricted toa finite domain. The results of simulations of 200 cells in aperiodic domain are presented in figure 2 at four differentdensities (see supplementary material, movies S1 and S2(available from stacks.iop.org/PhysBio/10/046002/mmedia)).The density is calculated as the number of cells per unit areaof the domain. First we carried out simulations in the absenceof noise (σ = 0). As in previous models of flocking behaviour

3

Phys. Biol. 10 (2013) 046002 L Coburn et al

(a) (b)

(c) (d )

Figure 2. (a) The order parameter, va versus time for 200 cells at different densities with noise, σ = 0. In each simulation the cells align in arandom direction. The transient time varies as the density is changed. (b)–(d) Snapshots of 200 cells at ρ = 0.33 and σ = 0 with randominitial positions and orientations. Inset: the order parameter versus time adjusted to the mean time between collisions. The curvesapproximately overlap. (b) t = 50, (c) t = 500, (d) t = 1500. The red arrows indicate the instantaneous velocity of each cell. A = 0.43,ω = 0.9, R0 = 0.5, γ = 0.1 and v0 = 1.

[9, 10] we characterize the collective alignment of the cellpopulation by the order parameter va defined as:

va =∣∣∑n

i=1 vi

∣∣∑ni=1 |vi| (5)

where vi is the velocity of the ith cell. The order parametervaries between 0 and 1 and measures the degree of collectivealignment of the cells. Initially cells are given random positionand orientation.

As shown in figure 2 in the absence of noise the cellsgradually form a perfectly aligned group moving in a randomlychosen common direction. Although asymptotically the orderparameter tends to unity in all simulations, the transient timenecessary to reach the ordered state increases as the celldensity is reduced. This is expected since at lower densitiesthe collisions are less frequent, thus the consensus directionis reached over a longer time since the local orientationspropagate more slowly through the group.

The ordered collective motion also develops in a circulardomain (see supplementary material, movie S3 (availablefrom stacks.iop.org/PhysBio/10/046002/mmedia)) where theperimeter of the region is made up of immotile cells thatthe moving cells interact with via CIL. In this case thecollective alignment is described by the group angularmomentum defined by:

vc = 1

n

∣∣∣∣∣n∑

i=1

sin(βi)

∣∣∣∣∣ (6)

where βi is the angle between the direction of motion of theith cell and the radial direction relative to the centre of thecircular domain. The tendency for motile cells in confinementto circulate collectively as a group has been described in [35].

3.2. Phase transition and shape dependence

Previous work on flocking models with non-local interactionshave shown that varying the strength of the stochasticity ofself-propelled point-like particles leads to a transition betweenorganized collective motion and random orientations [9, 35].This transition has similarities to the phase transition inferromagnets when the temperature passes through the criticalpoint. To test the sensitivity of organized collective movementin our model of cell migration to the strength of stochasticeffects we carried out a set of simulations with increasing noiselevels. Figure 3(a) presents the order parameter at stationarystate as a function of noise amplitude, σ , for four differentcell shapes. These results clearly indicate a transition similarto the self-propelled particle models, i.e. there is a sharp dropin collective alignment of cells as the noise level approachesa critical value above which the group can no longer align.Note, that the noise intensity at which the transition takes placedepends on the cell shape parameters, suggesting that certaincell shapes (i.e. protrusion distributions) have better ability forgenerating robust global alignment, than do others. We findthat broadly distributed protrusions, corresponding to largervalues of ω (as in figures 1(b) and (c)), in general, produce

4

Phys. Biol. 10 (2013) 046002 L Coburn et al

(a) (b)

(c) (d )

Figure 3. (a) The order parameter, va in the stationary state versus noise intensity σ for four different cell shapes in periodic domain withρ = 0.32. As the noise is increased there is a sharp transition from collective alignment to random movement. (b) The order parameter va inthe stationary state versus density ρ where σ = 0.7. Alignment improves as density is increased. �: ω = 1.5 and A = 0.32 (red), O: ω = 0.9and A = 0.44 (blue), ♦: ω = 0.5 and A = 0.56 (magenta), �: ω = 0.3 and A = 0.74 (green). (c) va versus A protrusion length,ω = 0.9, σ = 0.8. � (cyan): ρ = 0.8; O (red): ρ = 0.64; � (green): ρ = 0.45; ♦ (magenta): ρ = 0.33; (blue): ρ = 0.25. (d) va versus vo cellspeed, A = 0.44, ω = 0.9, σ = 0.8. O (blue): ρ = 0.64; � (red): ρ = 0.45; ♦ (green): ρ = 0.33; � (cyan): ρ = 0.25.

better alignment for a given level of noise, than does havingstrongly polarized cell shapes with protrusions concentrated atthe front of the cell even though such cell shapes yield fastermoving cells (figure 1(a)). This appears to be a consequence ofthe need to balance cell polarization and directional motilitywith the need for social cell–cell interactions as sensed bylateral protrusions, in order to maintain the collective directionin the presence of noise.

A similar, but more gradual, transition also takes placewhen the cell density is varied at a fixed value of σ ,i.e. higher density corresponds to stronger alignment (seefigure 3(b)). This is again in agreement with particle basedflocking models, as well as experimental data on swarminglocusts [36], suggesting that this cell model belongs to thesame universality class, in spite of the absence of non-localinteractions. To further investigate the shape dependence ofthe collective motion figure 3(c) show the order parameteras a function of maximum protrusion length A while thewidth ω is kept constant at different cell densities. The resultsshow that although a stronger polarization enhances global

alignment any polarized cells form an ordered state when thedensity is sufficiently high. Note at high density and highprotrusion length va drops as there is no longer the free spacefor protrusions to grow into and CIL prevents the cells frommoving. Figure 3(d) shows the order parameter as a function ofthe cell speed parameter illustrating that a stronger alignmentis reached when the cells move faster, i.e. when the frequencyof cell collisions increases.

Recent results based on a self-propelled particle modelwith only pairwise short-range interaction [37] suggest thatcollective alignment can arise in general from inelasticcollisions, i.e. when the relative angle between the orientationof two particles is reduced as a result of their interaction. Sincethe cells in our model are represented by spatially extendedcontours, in this case there is no unique relationship betweenthe orientations of two cells before and after their interactions.However, we can characterize the cell–cell interaction byaveraging over the possible pairwise collisions.

To investigate how the collective alignment depends onthe cell shape we carried out simulations of pairwise collisions

5

Phys. Biol. 10 (2013) 046002 L Coburn et al

(a) (b)

Figure 4. (a) The order parameter of groups of 200 cells with shape parameters shown in table 1. In the presence of noise, σ = 0.7 anddensity, ρ = 0.32. The shapes corresponding to lower values of e produce stronger alignment. (b) Stationary state order parameter versus efor many different cell shapes in the range A = 0.3–0.9, ω = 0.3–1.65 with noise, σ = 0.7.

Table 1. Restitution coefficient of pairwise collisions, e andstationary state order parameter, va (where n = 200, ρ = 0.32 andσ = 0.7) measured for different cell shapes (see figure 4).

ω A e va

(a) 1.5 0.3 0.47 0.77(b) 1.2 0.3 0.52 0.72(c) 0.9 0.3 0.58 0.61(d) 0.6 0.5 0.65 0.52(e) 0.3 0.7 0.69 0.18

in which the relative angle between the orientation of twocells before collision, θi, was varied from 0 to π in smallincrements so that the collisions are initially glancing andfinally head on. At each choice of initial relative angle aset of collisions are carried out varying the initial startingposition so that the point of contact ranges from the rear to thefront of the cells. We measured the final relative angles, θ f ,and calculated its average value over the number of collisionscarried out at each θi, to obtain the average final relative angleafter collision: 〈 θ f | θi〉 = f ( θi). We define the restitutioncoefficient of cell collisions as the ratio between the averagefinal and initial angles:

e = 1

M

M∑j=1

〈 θ f | θi = π j/M〉π j/M

. (7)

This quantity provides a statistical measure of the averagetendency to reduce the relative angle between two cells inpairwise collisions. The values obtained for different cellshapes are shown in table 1. Smaller values correspond tostronger average pairwise alignment that are expected toincrease the velocity correlations in a cell population. Thisis confirmed in figure 4 where we show the order parameterof groups of cells in the presence of noise for the same cellshapes. Comparing the values of e and the corresponding orderparameters in the stationary state shows that cell shapes thatproduce stronger average alignment in pairwise collisions alsocreate more coherent collective migration in a cell population.

3.3. Collective chemotaxis

In this section we investigate the role of the previouslydescribed collective motion in enhancing the overall

chemotactic response of the cell population, a property thathas been observed in recent experimental studies [32, 33].Since here we focus on the collective behaviour we do notrepresent explicitly the processes that generate the chemotacticresponse at the level of individual cells. (For a recent modelthat describes the effect of chemoattractant on cell protrusionsand motility in single cells see [17].) Instead we model theoverall chemotactic response of cells by assuming that theyturn towards the direction of a fixed gradient, so that in betweencollisions the orientation of a freely moving cell follows theequation:

dθ0

dt= −α sin(θ0 − θc) + η(t) (8)

where α is the maximum turning rate due to chemotaxis,θc is the direction of the gradient and η(t) is a white noise(as in equation (1) above) representing the combined effectsof stochastic cell movement and errors from detecting thegradient.

Following [12, 13], we characterize the ability of cells tonavigate in the direction of the gradient by the chemotacticefficiency, λ, calculated as the normalized projection of theirdisplacement in the direction θc, defined as:

λ = 〈cos(θ0(t) − θc)〉 (9)

where 〈. . .〉 represents a time average. The maximum value ofthe chemotactic efficiency is λ = 1 when the cell is alignedperfectly in the direction of the gradient. For single cells λ

increases with the turning rate α and decreases when thenoise intensity is increased. The chemotactic efficiency of cellgroups is calculated by averaging the efficiency of individualcells: � = ∑n

i=1 λi/n, where n is the number of cells and λi

is the chemotactic efficiency of the ith cell calculated from (9)above.

Simulations were performed for various cell densities andfor different values of the chemotactic turning rate α. Resultsof the simulations (figure 5(a)) show that in all cases after acertain transient time the chemotactic efficiency of the groupreaches approximately the same value as the order parameterof the same cell population without an external gradient. Thusthe chemotactic efficiency of the group is determined by theability of cells to coordinate their direction of movement,that depends on density and cell shape, but it is essentially

6

Phys. Biol. 10 (2013) 046002 L Coburn et al

(a) (b)

Figure 5. (a) Chemotactic efficiency, � versus density of populations of 200 cells migrating toward a chemo-attractant with differentturning rates α = 0.012 (magenta �), 0.015 (green �), 0.018 (blue O) and 0.021 (red ♦) with σ = 0.7 compared with the orderparameter in the absence of chemo-attractant (black solid line). The horizontal lines show the chemotactic efficiency of single cells forcomparison, for the same values of α and σ . (b) Chemotactic efficiency of 100 cells versus proportion of leaders, κ at density, ρ = 0.05(magenta �), 0.1 (green �), 0.2 (blue O) and 0.3 (red ♦) with σ = 0.7. Note that a very small proportion of leader cells, that maintaintheir orientation along the gradient, can produce near maximum chemotactic efficiency in the group (see supplementary movie S4 (availablefrom stacks.iop.org/PhysBio/10/046002/mmedia)).

independent of the ability of cells to turn in the directionof the gradient. This is because, non-chemotactic cells alignthemselves spontaneously in an arbitrary direction, and anysmall bias acting at the level of individual cells is sufficient toturn such ordered group into the direction of the gradient. Theonly consequence of the turning rate α is that the time neededto reach the direction of the gradient is longer when the groupis composed of cells with weaker chemotactic response.

Figure 5(a) shows that the collective migration leadsto a strong enhancement of the chemotactic efficiency of agroup with respect to single cells, in agreement with theexperimental observations [32]. Note, that the same typeof collective chemotaxis also appears in heterogeneous cellpopulations, where only a very small proportion of ‘leadercells’, that have the ability to sense the chemical gradient,are sufficient to steer the group in the direction of thechemoattractant (see figure 5(b)), and supplementary movie S4(available from stacks.iop.org/PhysBio/10/046002/mmedia)).Thus the underlying newly observed phenomena of collectivechemotaxis in cell populations is in fact closely related to theone described earlier for migrating animal groups guided by asmall proportion of leaders [11–13].

4. Discussion

Many biological systems consist of individuals interactingwith one another and responding to the conditions of theirenvironment. Often these processes are interrelated and it isthe collective behaviour of the system that determines theeffectiveness of the response. This form of emergent responseto environmental information is most commonly studied inwhole organisms, for example schools of fish that respondto the approach of a predator [38] or migratory species that

track a resource gradient. However, it is increasingly becomingevident that underlying principles exist that extend acrossmultiple scales and taxa, from interacting animal populationsto the functioning of cells.

In this work we have introduced a simple, yet realisticmodel of cellular interaction, inspired by experimental workon the phenomena of CIL. Our model represents thebehaviour of individual cells from a mechanistic perspective;collisions induce a reflexive reshaping of cell protrusionsand a consequent change in orientation. We have shown thatdepending on the cell shape such interaction may lead toinelastic collisions that produce increased alignment. Thusthe contact based cell interaction is analogous with thealignment rule of the particle based flocking models [9, 10].Consequently, at the collective level, pairwise interactionslead to global cohesion, hence CIL leads to behaviour thatis qualitatively equivalent to that observed in classic models ofanimal grouping, despite interaction ranges being restricted tothe extent of individual cells. For cell populations interactingvia CIL, as in larger organisms, there is a phase transition fromdisorder to an ordered regime when local noise and overalldensity is varied. An example of such transition can be seenin neural crest cells where the chemokine Sdf1 increases theactivity of small GTPases and stabilises cell protrusions [32],effectively reducing the noise of the individual cell motilitythat induces coherent collective migration.

Our model also provides support for the use of explicitalignment rule in particle based models of neural crest cellmigration [26]. Recent studies have found that co-attractionbetween cells may play a role in collective migration [26, 27].This is also consistent with the framework of flocking models[9, 10]. While in the model that we considered here thedensity of cells was maintained by closed boundaries; in anunconstrained system attractive interactions are necessary to

7

Phys. Biol. 10 (2013) 046002 L Coburn et al

avoid dispersion. The strength of such chemo-attraction thenregulates the density of cells and therefore determines thecharacteristics of collective migration.

When cell functioning is dependent on chemotaxis thepossibility of spontaneous self-organization has importantimplications. In situations when cell density is high andenvironmental sensing is imperfect, it is aggregate quantitiesthat determine an individual cells ability to accurately respondto chemical gradients. For homogeneous populations, afterrelaxation of transients, the accuracy achieved by a singlecell is equivalent to the degree of collective cohesion. Thisquantity is affected by cell shape (through its influence onpairwise interactions) and overall cell density, but notably,not by the detection capacity of individual cells. Theaccuracy of chemotaxis is therefore determined by thecollective system and not by the properties of individuals inisolation.

Acknowledgments

We would like to thank V Lobaskin and R Mayor for usefulcomments and discussions.

References

[1] Montell D J 2008 Morphogenetic cell movements: diversityfrom modular mechanical properties Science 322 1502–5

[2] Rørth P 2009 Annu. Rev. Cell Dev. Biol. 25 407–29[3] Friedl P and Gilmour D 2009 Nature Rev. Mol. Cell Biol.

10 445–57[4] Mehes E, Mones E, Nemeth V and Vicsek T 2012 PLoS ONE

7 e31711[5] Kulesa P M and Fraser S E 1998 Dev. Biol. 204 327–44[6] Teddy J M and Kulesa P M 2004 Development 131 6141–51[7] Friedl P and Wolf K 2003 Nature Rev. Cancer 3 362–74[8] Deisboeck T S and Couzin I D 2009 BioEssays 31 190–7[9] Vicsek T, Czirok A, Ben-Jacob E, Cohen I and Shochet O

1995 Phys. Rev. Lett. 75 1226–9[10] Couzin I D, Krause J, James R, Ruxton G D and Franks N R

2002 J. Theor. Biol. 218 1–11[11] Couzin I D, Krause J, Franks N R and Levin A S 2005 Nature

433 513–6[12] Guttal V and Couzin I D 2010 Proc. Natl Acad. Sci.

107 16172–7

[13] Torney C, Levin S A and Couzin I D 2010 Proc. Natl Acad.Sci. 107 20394–9

[14] Ananthakrishnan R and Ehrlicher A 2007 Int. J. Biol. Sci.3 303–17

[15] Bosgraaf L and van Haastert P J M 2009 PLoS ONE 4 e5253[16] Insall R H 2010 Nature Rev. Mol. Cell Biol. 11 453–8[17] Neilson M P, Veltman D M, van Haastert P J M, Webb S D,

Mackenzie J A and Insall R H 2011 PLoS Biol. 9 e1000618[18] Trepat X, Wasserman M R, Angelini T E, Millet E, Weitz D A,

Butler J P and Fredberg J 2009 Nature Phys. 5 426–30[19] Bindschadler M and McGrath J 2007 J. Cell Sci. 120 876–84[20] Farhadifar R, Roper J-C, Aigouy B, Eaton S and Julicher F

2007 Curr. Biol. 17 2095–210[21] Puliafito A, Hufnagel L, Neveu P, Streichan S, Sigal A,

Fygenson D K and Shraiman B I 2012 Collective and singlecell behavior in epithelial contact inhibition Proc. NatlAcad. Sci. 109 739–44

[22] Graner F and Glazier J A 1992 Phys. Rev. Lett. 69 2013–6[23] Szabo A, Unnep R, Mehes E, Twal W O, Argraves W S,

Cao Y and Czirok A 2010 Collective cell motion inendothelial monolayers Phys. Biol. 7 046007

[24] Kabla A J 2012 J. R. Soc. Interface 9 3268–78[25] De Calisto J, Araya C, Marchant L, Riaz C F and Mayor R

2005 Development 132 2587–97[26] Carmona-Fontaine C, Theveneau E, Tzekou A, Tada A,

Woods M and Page K 2011 Dev. Cell 21 1026–37[27] McLennan R, Dyson L, Prather K, Morrison J, Baker R

and Maini P 2012 Development 139 2935–44[28] Abercrombie M and Heaysman J E M 1953 Exp. Cell Res.

5 111–31[29] Abercrombie M 1979 Nature 281 259–62[30] Carmona-Fontaine C, Matthews H K, Kuriyama S, Moreno M,

Dunn G A, Parsons M, Stern C D and Mayor R 2008 Nature456 957–61

[31] Mayor R and Carmona-Fontaine C 2010 Trends Cell Biol.20 319–28

[32] Theveneau E, Marchant L, Kuriyama S, Gull M, Moepps B,Parsons M and Mayor R 2010 Dev. Cell 19 39–53

[33] Theveneau E and Mayor R 2010 Small GTPases 1 113–7[34] Vedel S, Tay S, Johnston D M, Bruus H and Quake S R 2012

Proc. Natl Acad. Sci. 110 129–34[35] Szabo B, Szollosi G J, Gonci B, Juranyi Z, Selmeczi D

and Vicsek T 2006 Phys. Rev. E 74 061908[36] Buhl J, Sumpter D J T, Couzin I D, Hale J J, Despland E,

Miller E R and Simpson S J 2006 Science 312 5778[37] Grossman D, Aronson I S and Ben-Jacob E 2008 New J. Phys.

10 023036[38] Handegard N O, Boswell K M, Ioannou C C, Leblanc S P,

Tjostheim D B and Couzin I D 2012 Curr. Biol. 22 1213–7

8